Abstract. We investigate the problem of calibrating an exponential Lévy model based on market prices of vanilla options. We show that this inverse problem is in general severely ill-posed and we derive exact minimax rates of convergence. The estimation procedure we propose is based on the explicit inversion of the option price formula in the spectral domain and a cut-off scheme for high frequencies as regularisation.

We present a generic non-nested Monte Carlo procedure for computing true upper bounds for Bermudan products, given an approximation of the Snell envelope. The pleonastic "true" stresses that, by construction, the estimator is biased above the Snell envelope. The key idea is a regression estimator for the Doob martingale part of the approximative Snell envelope, which preserves the martingale property. The so constructed martingale can be employed for computing tight dual upper bounds without nested simulation. In general, this martingale can also be used as a control variate for simulation of conditional expectations. In this context, we develop a variance reduced version of the nested primal-dual estimator (Andersen and Broadie, 2004). Numerical experiments indicate the efficiency of the proposed algorithms.

In this paper we develop several regression algorithms for solving general stochastic optimal control problems via Monte Carlo. This type of algorithms is particulary useful for problems with high-dimensional state space and complex dependence structure of the underlying Markov process with respect to some control. The main idea of the algorithms is to simulate a set of trajectories under some reference measure P * and to use a dynamic program formulation combined with fast methods for approximating conditional expectations and functional optimizations on these trajectories. Theoretical properties of the presented algorithms are investigated and convergence to the optimal solution is proved under mild assumptions. Finally, we present numerical results showing the efficiency of regression algorithms in a case of a highdimensional Bermudan basket options, in a model with a large investor and transaction costs.

Bermudan options, Nonparametric regression, Boundary condition, Suboptimal stopping rules, 62G08, 65C05, 60G40, G10, G12, G13,

In this article, the problem of semi-parametric inference on the parameters of a multidimensional Lévy process Lt with independent components based on the low-frequency observations of the corresponding time-changed Lévy process L T (t) , where T is a nonnegative, nondecreasing real-valued process independent of Lt, is studied. We show that this problem is closely related to the problem of composite function estimation that has recently gotten much attention in statistical literature. Under suitable identifiability conditions, we propose a consistent estimate for the Lévy density of Lt and derive the uniform as well as the pointwise convergence rates of the estimate proposed. Moreover, we prove that the rates obtained are optimal in a minimax sense over suitable classes of time-changed Lévy models. Finally, we present a simulation study showing the performance of our estimation algorithm in the case of time-changed Normal Inverse Gaussian (NIG) Lévy processes.